Transactions of the American Mathematical Society

Author(s): Uwe Nagel
Journal: Trans. Amer. Math. Soc. 351 (1999), 4381-4409.
MSC (1991): Primary 14M05; Secondary 13H10
Posted: April 20, 1999
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Abstract: In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.

1.: E. Ballico, On singular curves in positive characteristic, Math. Nachr. 141 (1989), 267-273. MR 90h:14042
2.: M. Brodmann, U. Nagel, Bounding cohomological Hilbert functions by hyperplane sections, J. Algebra 174 (1995), 323-348. MR 96d:14017
3.: D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, 1995. MR 97a:13001
4.: D. Eisenbud, S. Gôto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89-133. MR 85f:13023
5.: D. Eisenbud, J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom. 1 (1992), 15-30. MR 92i:14035
6.: D. Eisenbud, J. Harris, On varieties of minimal degree (A centennial account), Proceedings of Symposia in Pure Mathematics, Vol. 46 (1987), 3-13. MR 89f:14042
7.: D. Eisenbud, J. Koh, Some linear syzygy conjectures, Adv. Math. 90 (1991), 47-76. MR 93e:13019
8.: E. G. Evans, P. Griffith, The syzygy problem, Ann. Math. 114 (1981), 323-333. MR 83i:13006
9.: L. Gruson, R. Lazarsfeld, C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491-506. MR 85g:14033
10.: J. Harris, The genus of space curves, Math. Ann 249 (1980), 191-204. MR 81i:14022
11.: J. Harris, A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 36-68. MR 82h:14010
12.: J. Harris, D. Eisenbud, Curves in projective space, Chapter 3, Les Presses de l'Universite, Montreal, 1982. MR 84g:14024
13.: R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer-Verlag, 1977. MR 57:3116
14.: C. Huneke, B. Ulrich, General hyperplane sections of algebraic varieties, J. Algebraic Geom. 2 (1993), 487-505. MR 94b:14046
15.: S. Lvovski, On graded Betti numbers for finite subsets of curves, Preprint, Bonn 1997.
16.: D. Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Studies 59, Princeton Univ.Press, 1966. MR 35:187
17.: U. Nagel, On the defining equations and syzygies of arithmetically Cohen-Macaulay varieties in arbitrary characteristic, J. Algebra 175 (1995), 359-372. MR 96f:13023
18.: U. Nagel, On arithmetically Buchsbaum subschemes and liaison, Habilitationsschrift, Paderborn 1996.
19.: U. Nagel, Y. Pitteloud, On graded Betti numbers and geometrical properties of projective varieties, Manuscripta Math. 84 (1994), 291-314. MR 96d:13018
20.: U. Nagel, P. Schenzel, Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity, Nagoya Math. J. (to appear).
21.: U. Nagel, W. Vogel, Bounds for Castelnuovo's regularity and the genus of projective varieties, In: Topics in Algebra. Banach Center Publications, Vol. 26, Part 2, 163-183, PWN-Polish Scientific Publishers, Warsaw 1990. MR 93k:13024
22.: J.-P. Serre, Faiscaux algébriques cohérents, Ann. Math. 61 (1955), 197-278. MR 16:953c
23.: J. Stückrad, W. Vogel, Castelnuovo bounds for certain subvarieties in $\mathbb{ P}^n$ , Math. Ann. 276 (1987), 341-352. MR 88e:13013
24.: J. Stückrad, W. Vogel, Buchsbaum rings and applications, Springer-Verlag, 1986. MR 88h:13011a
25.: R. Treger, On equations defining arithmetically Cohen-Macaulay schemes, II, Duke Math. J. 48 (1981), 35-47. MR 84j:14050b
26.: N. V. Trung, G. Valla, Degree bounds for the defining equations of arithmetically Cohen-Macaulay varieties, Math. Ann. 281 (1988), 209-218. MR 89k:14083

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14M05, 13H10

Uwe Nagel
Affiliation: Fachbereich Mathematik und Informatik, Universität-Gesamthochschule Paderborn, D--33095 Paderborn, Germany
Email: uwen@uni-paderborn.de

DOI: 10.1090/S0002-9947-99-02357-0
PII: S 0002-9947(99)02357-0
Keywords: Minimal generator, local cohomology, Castelnuovo-Mumford regularity, arithmetically Buchsbaum scheme, rational normal scroll
Received by editor(s): August 27, 1997
Posted: April 20, 1999
Additional Notes: The material of this paper is part of the author's Habilitationsschrift [18].
Copyright of article: Copyright 1999, American Mathematical Society

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